## Section9.6Chapter 9 Summary and Review

### Subsection9.6.1Glossary

• horizontal

• vertical

• parallel

• perpendicular

• conic section

• circle

• ellipse

• parabola

• hyperbola

• central conic

• major axis

• minor axis

• vertices

• covertices

• transverse axis

• conjugate axis

• asymptote

• branch

### Subsection9.6.2Key Concepts

1. #### Horizontal and Vertical Lines.

1. The equation of the horizontal line passing through $$(0, b)$$ is

\begin{gather*} y=b \end{gather*}
2. The equation of the vertical line passing through $$(a,0)$$ is

\begin{gather*} x=a \end{gather*}

2. #### Slopes of Horizontal and Vertical Lines.

The slope of a horizontal line is zero.

The slope of a vertical line is undefined.

3. #### Parallel and Perpendicular Lines.

1. Two lines are parallel if their slopes are equal, that is, if

\begin{gather*} m_1 = m_2 \end{gather*}

or if both lines are vertical.

2. Two lines are perpendicular if the product of their slopes is $$-1\text{,}$$ that is, if

\begin{gather*} m_1m_2 = -1 \end{gather*}

or if one of the lines is horizontal and one is vertical.

4. The distance between two points is the length of the segment joining them.

5. #### Distance Formula.

The distance $$d$$ between points $$P_1(x_1, y_1)$$ and $$P_2(x_2, y_2)$$ is

\begin{gather*} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \end{gather*}

6. The midpoint of a segment is the point halfway between its endpoints, so that the distance from the midpoint to either endpoint is the same.

7. #### Midpoint Formula.

The midpoint of the line segment joining the points $$P_1(x_1, y_1)$$ and $$P_2(x_2, y_2)$$ is the point $$M(\overline{x},\overline{y})$$ where

\begin{gather*} \overline{x}=\frac{x_1+x_2}{2} \quad\text{ and } \overline{y}=\frac{y_1+y_2}{2} \end{gather*}

8. A circle is the set of all points in a plane that lie at a given distance, called the radius, from a fixed point called the center.

9. #### Circle.

The equation for a circle of radius $$r$$ centered at the point $$(h, k)$$ is

\begin{gather*} (x-h)^2+(y-k)^2=r^2 \end{gather*}

10. #### Ellipse.

The standard form for an ellipse centered at $$(h, k)$$ is

\begin{gather*} \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \end{gather*}

11. #### Hyperbola.

The equation for a hyperbola centered at the point $$(h, k)$$ has one of the two standard forms:

\begin{gather*} \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \end{gather*}
\begin{gather*} \frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1 \end{gather*}

12. #### Conic Sections.

The graph of $$Ax^2+Cy^2+Dx+Ey+F=0$$ is

1. a circle of $$A=C\text{.}$$

2. a parabola if $$A=0$$ or $$C=0$$ (but not both).

3. an ellipse if $$A$$ and $$C$$ have the same sign.

4. a hyperbola if $$A$$ and $$C$$ have opposite signs.

13. A system of two quadratic equations may have up to four solutions.

### Exercises9.6.3Chapter 9 Review Problems

#### Exercise Group.

For Problems 1 and 2, decide whether the lines are parallel, perpendicular, or neither.

##### 1.

$$y=\dfrac{1}{2}x+3 \text{;}$$ $$\quad x-2y=8$$

parallel

##### 2.

$$4x-y=6 \text{;}$$ $$\quad x+4y=-2$$

perpendicular

#### 3.

Write an equation for the line that is parallel to the graph of $$2x + 3y = 6$$ and passes through the point $$(1, 4)$$

$$y=\dfrac{-2}{3}x+\dfrac{14}{3}$$

#### 4.

Write an equation for the line that is perpendicular to the graph of $$2x + 3y = 6$$ and passes through the point $$(1, 4)$$

$$y=\dfrac{3}{2}x+\dfrac{5}{2}$$

#### 5.

Two vertices of the rectangle $$ABCD$$ are $$A(3, 2)$$ and $$B(7, −4)\text{.}$$ Find an equation for the line that includes side $$\overline{BC}\text{.}$$

$$y=\dfrac{2}{3}x-\dfrac{26}{3}$$

#### 6.

One leg of the right triangle $$PQR$$ has vertices $$P(-8, -1)$$ and $$Q(-2, -5)\text{.}$$ Find an equation for the line that includes the leg $$\overline{QR} \text{.}$$

$$y=\dfrac{3}{2}x-2$$

#### 7.

Find the perimeter of the triangle with vertices $$A(-1, 2)\text{,}$$ $$B(5, 4)\text{,}$$ $$C(1, -4)\text{.}$$ Is $$\Delta ABC$$ a right triangle?

21.59; yes

#### 8.

Find the midpoint of each side of $$\Delta ABC$$ from the previous problem. Join the midpoints to form a new triangle, and find its perimeter.

10.8

#### Exercise Group.

For Problems 9–18, graph the conic section.

##### 9.

$$x^2+y^2=9$$

##### 10.

$$\dfrac{x^2}{9}+y^2=1$$

##### 11.

$$\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$$

##### 12.

$$\dfrac{y^2}{6}-\dfrac{x^2}{8}=1$$

##### 13.

$$4(y-2)=(x+3)^2$$

##### 14.

$$(x-2)^2+(y+3)^2=16$$

##### 15.

$$\dfrac{(x-2)^2}{4}+\dfrac{(y+3)^2}{9}=1$$

##### 16.

$$\dfrac{(x+4)^2}{12}+\dfrac{(y-2)^2}{6}=1$$

##### 17.

$$\dfrac{(x-2)^2}{4}+\dfrac{(y+3)^2}{9}=1$$

##### 18.

$$(x-2)^2+4y=4$$

#### Exercise Group.

For Problems 19–30

1. Write the equation of each conic section in standard form.

2. Identify the conic and describe its main features.

##### 19.

$$x^2+y^2-4x+2y-4=0$$

1. $$\displaystyle (x-2)^2+(y+1)^2=9$$

2. Circle: center $$(2,-1)\text{,}$$ radius 3

##### 20.

$$x^2+y^2-6y-4=0$$

1. $$\displaystyle x^2+(y-3)^2=13$$

2. Circle: center $$(0,3)\text{,}$$ radius $$\sqrt{13}$$

##### 21.

$$4x^2+y^2-16x+4y+4=0$$

1. $$\displaystyle \dfrac{(x-2)^2}{4}+\dfrac{(y+2)^2}{16}=1$$

2. Ellipse: center $$(2,-2)\text{,}$$ $$a=2\text{,}$$ $$b=4$$

##### 22.

$$8x^2+5y^2+16x-20y-12=0$$

1. $$\displaystyle \dfrac{(x+1)^2}{5}+\dfrac{(y-2)^2}{8}=1$$

2. Ellipse: center $$(-1,2)\text{,}$$ $$a=\sqrt{5}\text{,}$$ $$b=\sqrt{8}$$

##### 23.

$$x^2-8x-y+6=0$$

1. $$\displaystyle y+10=(x-4)^2$$

2. Parabola: vertex $$(4,10)\text{,}$$ opens upward, $$a=1$$

##### 24.

$$y^2+6y+4x+1=0$$

1. $$\displaystyle x-2=\dfrac{-1}{4}(y+3)^2$$

2. Parabola: vertex $$(2,-3)\text{,}$$ opens left, $$a=\dfrac{-1}{4}$$

##### 25.

$$x^2+y=4x-6$$

1. $$\displaystyle y+2=-(x-2)^2$$

2. Parabola: vertex $$(2,-2)\text{,}$$ opens downward, $$a=-1$$

##### 26.

$$y^2=2y+2x+2$$

1. $$\displaystyle x+\dfrac{3}{2}=\dfrac{1}{2}(y-1)^2$$

2. Parabola: vertex $$\left(\dfrac{-3}{2},1\right)\text{,}$$ opens right, $$a=\dfrac{1}{2}$$

##### 27.

$$2y^2-3x^2-16y-12x+8=0$$

1. $$\displaystyle \dfrac{(y-4)^2}{6}-\dfrac{(x+2)^2}{4}=1$$

2. Hyperbola: center $$(-2,4)\text{,}$$ transverse axis vertical, $$a=2\text{,}$$ $$b=\sqrt{6}$$

##### 28.

$$9x^2-4y^2-72x-24y+72=0$$

1. $$\displaystyle \dfrac{(x-4)^2}{4}-\dfrac{(y+3)^2}{9}=1$$

2. Hyperbola: center $$(4,-3)\text{,}$$ transverse axis horizontal, $$a=2\text{,}$$ $$b=3$$

##### 29.

$$2x^2-y^2+6y-19=0$$

1. $$\displaystyle \dfrac{x^2}{5}-\dfrac{(y-3)^2}{10}=1$$

2. Hyperbola: center $$(0,3)\text{,}$$ transverse axis horizontal, $$a=\sqrt{5}\text{,}$$ $$b=\sqrt{10}$$

##### 30.

$$4y^2-x^2+8x-28=0$$

1. $$\displaystyle \dfrac{y^2}{3}-\dfrac{(x-4)^2}{12}=1$$

2. Hyperbola: center $$(4,0)\text{,}$$ transverse axis vertical, $$a=2\sqrt{3}\text{,}$$ $$b=\sqrt{310}$$

#### 31.

An ellipse centered at the origin has a vertical major axis of length 16 and a horizontal minor axis of length 10.

1. Find the equation of the ellipse.

2. What are the values of $$y$$ when $$x = 4\text{?}$$

1. $$\displaystyle \dfrac{x^2}{25}+\dfrac{y^2}{64}=1$$

2. $$\displaystyle \dfrac{\pm 24}{5}$$

#### 32.

An ellipse centered at the origin has a horizontal major axis of length 26 and a vertical minor axis of length 18.

1. Find the equation of the ellipse.

2. What are the values of $$y$$ when $$x = 12\text{?}$$

1. $$\displaystyle \dfrac{x^2}{169}+\dfrac{y^2}{81}=1$$

2. $$\displaystyle \dfrac{\pm 45}{13}$$

#### Exercise Group.

For Problems 33–38, write an equation for the conic section with the given properties.

##### 33.

Circle: center at $$(-4, 3)\text{,}$$ radius $$2\sqrt{5}$$

$$(x+4)^2+(y-3)^2=20$$

##### 34.

Circle: endpoints of a diameter at $$(-5, 2)$$ and $$(1, 6)$$

$$(x+2)^2+(y-4)^2=13$$

##### 35.

Ellipse: center at $$(-1, 4)\text{,}$$ $$a = 4\text{,}$$ $$b = 2$$

$$\dfrac{(x+1)^2}{16}+\dfrac{(y-4)^2}{4}=1$$

##### 36.

Ellipse: vertices at $$(3, 6)$$ and $$(3, -4)\text{,}$$ covertices at $$(1, 1)$$ and $$(5, 1)$$

$$\dfrac{(x-3)^2}{4}+\dfrac{(y-1)^2}{25}=1$$

##### 37.

Hyperbola: center $$(2, -3)\text{,}$$ $$a = 4\text{,}$$ $$b = 3\text{,}$$ transverse axis horizontal

$$\dfrac{(x-2)^2}{16}-\dfrac{(y+3)^2}{9}=1$$

##### 38.

Hyperbola: one vertex at $$(−4, 1)\text{,}$$ one end of the vertical conjugate axis at $$(-3, 4)$$

$$(x+3)^2 -\dfrac{(y-1)^2}{9}=1$$

#### Exercise Group.

For Problems 39–42, graph the system of equations and state the solutions of the system

##### 39.

\begin{aligned}[t] 4x^2+y^2=25\\ x^2-y^2=-5 \end{aligned}

$$(\pm 2, \pm 3)$$

##### 40.

\begin{aligned}[t] 4x^2-9y^2+132=0\\ x^2+4y^2-67=0 \end{aligned}

$$(\pm\sqrt{3}, \pm 4)$$

##### 41.

\begin{aligned}[t] x^2+3y^2=13\\ xy=-2 \end{aligned}

$$(1,-2) \text{,}$$ $$(-1,2) \text{,}$$ $$\left(2\sqrt{3}, \dfrac{-1}{\sqrt{3}} \right) \text{,}$$ $$\left(-2\sqrt{3}, \dfrac{1}{\sqrt{3}} \right)$$

##### 42.

\begin{aligned}[t] x^2+y^2=17\\ 2xy=-17 \end{aligned}

$$\left(\dfrac{\sqrt{34}}{2}, \dfrac{\sqrt{34}}{2} \right) \text{,}$$ $$\left(\dfrac{-\sqrt{34}}{2}, \dfrac{-\sqrt{34}}{2} \right)$$

#### Exercise Group.

For Problems 43–48, write and solve an equation or a system of equations

##### 43.

Moia drives 180 miles in the same time that Fran drives 200 miles. Find the speed of each if Fran drives 5 miles per hour faster than Moia.

Moia: 45 mph, Fran: 50 mph

##### 44.

The perimeter of a rectangle is 26 inches and the area is 12 square inches. Find the dimensions of the rectangle.

12 in by 1 in

##### 45.

The perimeter of a rectangle is 34 centimeters and the area is 70 square centimeters. Find the dimensions of the rectangle.

7 cm by 10 cm

##### 46.

A rectangle has a perimeter of 18 feet. If the length is decreased by 5 feet and the width is increased by 12 feet, the area is doubled. Find the dimensions of the original rectangle.

7 ft by 2 ft

##### 47.

Norm takes a commuter train 10 miles to his job in the city. The evening train returns him home at a rate 10 miles per hour faster than the morning traintakes him to work. If Norm spends a total of 50 minutes per day commuting, what is the rate of each train?

Hattie's annual income from an investment is $32. If she had invested$200 more and the rate had been 1/2% less, her annual income would have been $35. What are the amount and rate of Hattie's investment? Answer. Amount:$800, rate: 4%